Rise Over Run

Definition

Rise over run is a way to express the slope or steepness of a line. It represents the change in the vertical direction (rise) divided by the change in the horizontal direction (run) between two points on a line. This concept is fundamental to graphing linear equations and finding the equation of a line.

5 Must Know Facts For Your Next Test

1. The rise over run of a line can be calculated by taking the difference in the $y$-coordinates of two points on the line and dividing it by the difference in the $x$-coordinates of those same two points.
2. The rise over run of a line is equivalent to the slope of the line, which determines the angle of the line relative to the $x$-axis.
3. A positive rise over run indicates a line that is sloping upward from left to right, while a negative rise over run indicates a line that is sloping downward from left to right.
4. The rise over run is a key component in both the slope-intercept form and the point-slope form of a linear equation.
5. Understanding the concept of rise over run is essential for graphing linear equations and determining the equation of a line given specific information about the line.

Review Questions

• Explain how the rise over run is used to determine the slope of a line.
  • The rise over run is directly equivalent to the slope of a line. To calculate the slope, you take the difference in the $y$-coordinates of two points on the line (the rise) and divide it by the difference in the $x$-coordinates of those same two points (the run). This ratio represents the steepness or incline of the line, with a positive value indicating an upward slope and a negative value indicating a downward slope.
• Describe how the rise over run is used in the slope-intercept form of a linear equation.
  • In the slope-intercept form of a linear equation, $y = mx + b$, the slope of the line is represented by the coefficient $m$. This $m$ value is directly equal to the rise over run of the line, as it represents the change in the $y$-direction divided by the change in the $x$-direction between any two points on the line. Understanding the relationship between the rise over run and the slope-intercept form is crucial for writing the equation of a line given information about its slope.
• Analyze how the rise over run can be used to find the equation of a line given a specific point on the line and the slope of the line.
  • When given a point $(x_1, y_1)$ that lies on a line and the slope (rise over run) $m$ of that line, you can use the point-slope form of a linear equation, $y - y_1 = m(x - x_1)$, to find the equation of the line. In this form, the slope $m$ represents the rise over run, which captures the steepness of the line. By substituting the known point and slope into the point-slope equation, you can solve for the $y$-intercept $b$ and write the full equation of the line in slope-intercept form, $y = mx + b$.

Related terms

Slope: The slope of a line is a measure of its steepness and is calculated as the change in the vertical direction (rise) divided by the change in the horizontal direction (run) between two points on the line.

Slope-Intercept Form: The slope-intercept form of a linear equation is $y = mx + b$, where $m$ represents the slope (rise over run) and $b$ represents the $y$-intercept of the line.

Point-Slope Form: The point-slope form of a linear equation is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a known point on the line and $m$ represents the slope (rise over run).